Step 1: Understanding the Concept:
The speed of an electromagnetic wave in a medium is given by \( v = \frac{c}{\sqrt{\mu_r \epsilon_r}} \), where \( c \) is the speed of light in a vacuum. Once \( v \) is found, the wavelength \( \lambda \) can be calculated using \( v = f\lambda \).
Step 2: Key Formula or Approach:
1. Refractive index \( n = \sqrt{\mu_r \epsilon_r} \)
2. Velocity in medium \( v = \frac{3 \times 10^8}{n} \)
3. Wavelength \( \lambda = \frac{v}{f} \)
Step 3: Detailed Explanation:
Given:
Relative Permeability \( \mu_r = 200 \)
Dielectric Constant \( \epsilon_r = 8 \)
Frequency \( f = 100 \, \text{MHz} = 100 \times 10^6 \, \text{Hz} = 10^8 \, \text{Hz} \)
Calculate refractive index \( n \):
\[ n = \sqrt{\mu_r \epsilon_r} = \sqrt{200 \times 8} = \sqrt{1600} = 40 \]
Calculate velocity \( v \):
\[ v = \frac{3 \times 10^8}{40} = 0.75 \times 10^7 \, \text{m/s} \]
Calculate wavelength \( \lambda \):
\[ \lambda = \frac{v}{f} = \frac{0.75 \times 10^7}{10^8} = 0.75 \times 10^{-1} \, \text{m} \]
\[ \lambda = 0.075 \, \text{m} = 7.5 \, \text{cm} \]
Step 4: Final Answer:
The wavelength is 7.5 cm.